Chebyshev Multi-section Matching The Chebyshev transformer is optimum bandwidth to allow ripple within the passband response, and is known as equally.

Chebyshev Multi-section MatchingThe Chebyshev transformer is optimum bandwidth to allow

ripple within the passband response, and is known as equally ripple.

Larger bandwidth than that of binomial matching.

The Chebyshev characteristics

(5.60)or (5.59) as rewritten

be can spolynomial Chebyshev

matching,for cosseclet

);()(2)(

spolynomial Chebyshev

121

m

nnn

x

xTxTxxTxT

2

N N-1

2 2

0

0

-1 0

m 0

Chebyshev response ( ) (sec cos )

1where the last term is for even , for odd .

2

1(0) (sec )

(sec )

Let to evaluate

1 1sec cosh[ cosh ( )]

N

cos

jnN m

LN m

L N m

m

Lm

L

Ae T

N N

Z ZA T A

Z Z T

A

Z Z

Z Z

-1 0

m

0

1

1 ln( / )h[ cosh ( )]

N 2

4Fraction bandwidth 2

1More accurate formula ln (5.64b)

2

L

m

nn

n

Z Z

f

f

Z

Z

Chebyshev Transformer Design

Example5.10: Design a three-section Chebyshev transformer to match a 100 load to a 50 line, with m=0.05?

Solution

2 2 33

-1 0

m

-1

33

For 3

( ) (sec cos ) (sec cos )

0.05

1 ln( / )sec cosh[ cosh ( )]

N 2

1 ln(100 / 50)cosh[ cosh ( )] 1.408 44.7

3 2 0.05

Using (5.61) and (5.60c) for (omit )

c

jn jN m m

m

Lm

m

j

N

Ae T Ae T

A

Z Z

T e

30 0

31 m 1

3 0 2 1

os3 : 2 sec 0.0698

cos : 2 3 (sec sec ) 0.1037

From symmetry ;

m

m

A

A

1 0 0

1

2 1 1

2

3 2 0

3

Using (5.64b)

0 : ln ln 2 ln50 2(0.0698) 4.051

57.5

1: ln ln 2 ln57.5 2(0.1037) 4.259

70.7

2 : ln ln 2 ln 70.7 2(0.1037) 4.466

87.0

Fraction ban

n Z Z

Z

n Z Z

Z

n Z Z

Z

0

4 44.7dwidth 2 2 ( ) 1.01

180mf

f

Using table design for N=3 and ZL/Z0=2 can find coefficient as 1.1475, 1.4142, and 1.7429. So Z1=57.37, Z2=70.71, and Z3=87.15.

Tapered Lines MatchingThe line can be continuously tapered instead of discrete multiple sections to achieve broadband matching.

Changing the type of line can obtain different passband characteristics.

Relation between characteristic impedance

and reflection coefficient

l

Z

Z

dz

de

L

z

zj

2;

)ln(2

1)(

00

2

Three type of tapered line

will be considered here

1) Exponential

2)Triangular

3) Klopfenstein

Exponential TaperThe length (L)of line should be greater than /2(l>) to minimize the mismatch at low frequency.

L

Le

ZZ

dzedz

de

Z

Z

La

eZzZ

LjL

azL

z

zj

L

az

sin

2

)ln(

)(ln2

1)(

)ln(1

)(

0

0

2

0

0

Triangular TaperThe peaks of the triangular taper are lower than the corresponding peaks of the exponential case.

First zero occurs at l=2

2

0

2/for /ln)1)/(2/4(0

2/0for /ln)/(20

]2/

2/sin[)ln(

2

1)(

)(0

2

02

L

Le

Z

Z

eZ

eZzZ

LjL

LzLZZLzLz

LzZZLz

L

L

Klopfenstein TaperFor a maximum reflection coefficient specification in the passband, the Klopfenstein taper yields the shortest matching section (optimum sense).

The impedance taper has steps at z=0 and L, and so does not smoothly join the source and load impedances.

AZ

Z

ZZ

ZZA

ALe

A

AAx

xAxA

xI

xdyyA

yAIAxAx

LzAL

zA

AZZzZ

mL

L

L

Lj

x

L

cosh );ln(

2

1cosh

)(cos)(

1cosh),( ;

2),( ;0),0(

valuesspecial the withfunction Besselmodified theis )(

1;1

)1(),(),(

0);,12

(cosh

ln2

1)(ln

0

00

00

22

0

2

1

0 2

21

200

Example5.11: Design a triangular, exponential, and Klopfenstein tapers to match a 50 load to a 100 line?

Solution

Triangular taper

2

2/for 2/1ln)1)/(2/4(

2/0for 2/1ln)/(2

]2/

2/sin)[

2

1ln(

2

1)(

100

100)( 2

2

L

L

e

ezZ

LzLLzLz

LzLz

Exponential taper

L

LL

a

eZzZ az

sin

2

)21ln()(

)2

1ln(

1

)( 0

Klopfenstein taper

cosh

)(cos)(

)346.0

(cosh)(cosh

346.0)ln(2

1

22

0

101

00

00

A

AL

A

Z

Z

ZZ

ZZ

mm

L

L

L

Bode-Fano CriterionThe criterion gives a theoretical limit on the minimum

reflection magnitude (or optimum result) for an arbitrary matching network

The criterion provide the upper limit of performance to tradeoff among reflection coefficient, bandwidth, and network complexity.

For example, if the response ( as the left hand side of next page) is needed to be synthesized, its function is given by applied the criterion of parallel RC

1ln

1ln

1ln

0

RCdωdω

mm

For a given load, broader bandwidth , higher m.

m 0 unless =o. Thus a perfect match can be achieved only at a finite number of frequencies.

As R and/or C increases, the quality of the match ( and/or m) must decrease. Thus higher-Q circuits are intrinsically harder to match than are lower-Q circuits.

Chapter 6

Microwave Resonators

RLC Series Resonant Circuit

Microwave resonators are used in a variety of applications, including filters, oscillators, frequency meters, and tuned amplifiers.

The operation of microwave resonators is very similar to that of the lumped-element resonators (such as parallel and series RLC resonant circuits) of circuit theory.

RRIP

CjLjRI

IZVIP

CjLjRZ

loss

inin

in

resistor by the dissipated Power ;2

1

resonator todeliveredPower

);1

(2

12

1

2

1

1

2

2

2*

Vout

Vin

circuitresonant a of loss theoft measuremen

;factor Quality

)1

(

21

) occur( Resonance21

)(22

)(2

capacitor thein storedenergy electric Average ;1

4

1

inductor thein storedenergy magnetic Average ;4

1

02

22

2

2

2

loss

me

lossin

em

emlossinin

emlossin

e

m

P

WWQ

LCR

I

PZ

WW

I

WWjP

I

PZ

WWjPP

CC

IW

LLIW

)2)2())(((

;2

2

)()1

1(

0 where,Let

1

21

141

22

21

41

22

resonanceAt

0020

2

0

2

20

2

2

0

02

20

2

00

0

2

2

00

RQjRLjR

LjRLC

LjRZ

CRRI

CI

P

W

R

L

RI

LI

P

WQ

in

loss

e

loss

m

CLX

R

XQ

00

1or where

;

0

2

resonator theof bandwidth fractionalpower -half The

BW

RjRRZ

QBW

BWjRQRRQ

jRZ

in

in

2

1 When

2

0